Compact open topology and CW homotopy type

The class W of spaces having the homotopy type of a CW complex is not closed under formation of function spaces. In 1959, Milnor proved the fundamental theorem that, given a space YE W and a compact Hausdorff space X, the space Y-X of continuous functions X --> Y, endowed with the compact open topology, belongs to W. P.J. Kahn extended this in 1982, showing that Y-X is an element of W if X has finite n-skeleton and pi(k) (Y) = 0, k > n. Using a different approach, we obtain a further generalization and give interesting examples of function spaces Y-X is an element of W where X is an element of W is not homotopy equivalent to a finite complex, and Y is an element of W has infinitely many nontrivial homotopy groups. We also obtain information about some topological properties that are intimately related to CW homotopy type. As an application we solve a related problem concerning towers of fibrations between spaces of CW homotopy type. (C) 2002 Elsevier Science B.V. All rights reserved.